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Topic: Calculate the equilibrium concentration of CO2 in beer  (Read 1586 times)

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Offline Enthalpy

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #15 on: August 12, 2020, 07:34:06 AM »
[...] Now think about a bubble of CO2 1" below the surface of the water. [...]

That's why I distinguished a pseudo-equilibrium where gas is produced in the depth and its concentration limited by the formation of bubbles. This is what happened at lake Nyos for instance.
https://en.wikipedia.org/wiki/Lake_Nyos
There, the liquid's pressure does prevent the formation of bubbles, and it matters for defining a maximum gas concentration.

But this situation is not an equilibrium. It can happen only if the gas production at depth is faster than the diffusion through the liquid.

In a real equilibrium, diffusion suffices to establish the gas concentration in the liquid, and this equilibrium is nearly uniform. A tiny gradient would result from gravity acting on the density difference between the liquid and the molecules of dissolved gas - this difference is so small that it's not usable technologically. Only a centrifuge could do something.

With the uniform concentration of a true equilibrium, the subsurface defines the gas concentration, and it depends only the the gas pressure above the liquid.

==========

I expect fermentation to produce dioxide within the whole beer volume, so there would be more dioxide at depth if the production is quick enough. That's nothing obvious, because the situation is unstable: an eddy current may release bubbles as a solution parcel rises, and this parcel gets lighter and rises faster. Such instability would reequilibrate the dioxide concentration efficiently.

In any case, I do NOT call that an equilibrium.

Offline Corribus

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #16 on: August 12, 2020, 09:17:13 AM »
You can't have both equilibrium and concentration gradient at the same time (at least as long as we don't talk about some highly exotic cases).
You can if the chemical/physical environment has a static spatial variation. Gas solubility depends on hydrostatic pressure:

https://pubs.acs.org/doi/10.1021/j100886a005

And hydrostatic pressure depends on depth. So the gas solubility depends on depth. I imagine also for gasses like CO2 that react with water, the equilibrium constant is probably also dependent on hydrostatic pressure.

Such an effect is only relevant for large water columns, because hydrostatic pressure differences are small in normal laboratory situations. But these effects would be important for determining, say, carbon dioxide content of the ocean. So I think this qualifies as one of the 'highly exotic cases' that you mention.
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Offline mjc123

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #17 on: August 12, 2020, 11:25:54 AM »
That is interesting, and I accept the correction. However, there is only a small change in the gas equilibrium pressure (16%) for a hydrostatic pressure of 100 atm. Moreover, if I understand the first page correctly (I can't access the full article), water equilibrated with gas at 1 atm is subjected to hydrostatic pressure, and the equilibrium gas pressure increases, i.e. the solubility for a given gas pressure decreases. That is what I would expect qualitatively, at high pressure there will be less free volume, so less capacity to dissolve gas. What this means is that the Henry's Law constant decreases slightly with increasing hydrostatic pressure.

But, OP, let's be clear. Henry's Law does not say "at higher pressure, more gas particles are forced through the water/gas interface into the liquid phase and are dissolved by the liquid (this is henry's law)" - at least, not if you're interpreting that to mean that dissolved concentration is proportional to hydrostatic pressure. It says that the concentration of dissolved gas is proportional to the partial pressure of that gas. It's just that the constant of proportionality varies slightly with the hydrostatic pressure. You are not given the information in the question to deal with that, and I guess you're not intended to - at a pressure of 2 atm, the effect will be very small. What I guess you're expected to do, but is absolutely not correct, is to multiply the Henry's Law constant by the hydrostatic pressure. At least, that's what you did, and got the "right" (wrong) answer.

Offline Corribus

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #18 on: August 12, 2020, 12:56:21 PM »
@mjc

In fairness, it wasn't obvious to me either. I had to do a literature search. And yes, basically as the hydrostatic pressure increases, the equilibrium gas pressure increases. Although they don't mention solubility in the article, my expectation based on intuition is also that solubility would decrease at depth. As you say, the effect isn't enormous, mostly I imagine because fluids aren't very compressible.

This does seems to translate into a dependence of the Henry Law Constant and solubility as a function of hydrostatic pressure:

https://aslopubs.onlinelibrary.wiley.com/doi/abs/10.4319/lo.1961.6.3.0365

(don't know if you can access that or not)

I have copied pertinent data here from the first article if you are interested.
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Offline Borek

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #19 on: August 12, 2020, 01:24:26 PM »
Gas solubility depends on hydrostatic pressure:

Perhaps I am missing something, but it sounds to me like another way of saying "Henry's law is only an approximation" (which is quite normal, every chemical/physical law I can think of that states linear proportionality fails at some point). Still, interesting find, thank you.

Quote
So I think this qualifies as one of the 'highly exotic cases' that you mention.

I was actually thinking about isotope separation in gravitational field (very tall columns, very low temperatures), but I was at the same time sure there are other cases that will fit.
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Offline mjc123

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #20 on: August 12, 2020, 05:56:26 PM »
Thanks for that reference @Corribus. In the light of above discussions, it is interesting to read the paragraph beginning "It has been implied occasionally..." regarding the difference between solubility and critical concentration for bubble formation.

Offline MNIO

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #21 on: August 12, 2020, 08:32:06 PM »
You can't have both equilibrium and concentration gradient at the same time (at least as long as we don't talk about some highly exotic cases).

not true.  equilibrium means concentration doesn't change over time.  It does NOT mean uniform concentration.

Offline Enthalpy

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Re: Calculate the equilibrium concentration of CO2 in beer
« Reply #22 on: August 19, 2020, 08:03:25 PM »
You can't have both equilibrium and concentration gradient at the same time (at least as long as we don't talk about some highly exotic cases).

not true.  equilibrium means concentration doesn't change over time.  It does NOT mean uniform concentration.

What about diffusion? Under usual conditions, other actions don't prevent it.

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